Control Engineering Practice 10 (2002) 633–644

A linear parameter variant HN control design for an induction motor E. Prempaina,*, I. Postlethwaitea, A. Benchaibb a

Department of Engineering, Control & Instrumentation Research, University of Leicester, University Road, Leicester LE1 7RH, UK b Alstom Technology, 9 rue Ampere, Massy 91345, France Received 20 October 2000; accepted 7 December 2001

Abstract A robust controller for an induction motor is designed using HN control theory and input–output feedback linearization. Because of the special structure of the state–space equations governing the induction motor a linear parameter varying (LPV) feedback controller, scheduled with rotor speed, is used for the inner current loop. An LPV observer is also used to estimate the flux vector. The application is based on the non-linear model and tracking requirements of a recently published benchmark, which describes an experimental set-up at the Laboratory of Electrical Engineering, Paris. The proposed controller delivers high performance over the entire operating range of the induction motor and compares favourably with other published results. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Induction motor; Input–output linearization; State-feedback; Linear matrix inequalities; HN optimization; LPV gain scheduling control

Notation In : n
n identity matrix. L2 : the space of signals with finite energy. jjGjjN ¼ supuAL2 jjyjjL2 =jjujjL2 : the infinity norm of G where yðtÞ is the output of the system G for a given input uðtÞ: Fl ðP; KÞ: the lower linear fractional transformation of P and K; defined as Fl ðP; KÞ :¼ P11 þ P12 K ðI  P22 KÞ1 P21 where the matrix P is partitioned as
 P11 P12 P¼ P21 P22 Ty;r : the input/output map r-y:

1. Introduction Induction motors, due to their size, low cost and high reliability, are widely used in industry mainly to transform electrical energy into mechanical energy. However, induction motors are rarely used for high precision tasks because they are significantly more difficult to control *Corresponding author. Tel.: +44-116-252-2874; fax: +44-116-2522619. E-mail address: [email protected] (E. Prempain).

than d.c. motors. Nowadays, therefore, there is great interest in developing high performance control laws to make induction motor performance rival that of the d.c. motor in a number of high precision applications (e.g. robotic applications). Induction motors are theoretically challenging for control engineers because as dynamical systems they are highly non-linear, the flux which is to be controlled (and sometimes the rotor speed) is not available. Also, physical parameter uncertainties, such as the variation of the rotor resistance with temperature, affect significantly the dynamics of the system. Overviews of the important induction motor control techniques are given in Bose (1998) and Taylor (1994). In this paper, the design of a controller for an induction motor is based on a linear parameter variant (LPV) of HN control, the so-called LPV theory (Apkarian & Gahinet, 1995; Gahinet, Nemirovski, Laub, & Chilali, 1995), and input/output feedback linearization. It is assumed that only the stator currents and the rotor speed are available for measurement. Broadly speaking, the control law consists of a fast inner loop used to track stator current references which are generated by a non-linear input–output linearization state feedback gain. Realistic simulation results show that this control law is robust with respect to electrical parameter uncertainties, noise measurement and load torque variations. The proposed control law is easy to

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E. Prempain et al. / Control Engineering Practice 10 (2002) 633–644

tune and easily implemented due to the low orders of the estimator and the feedback controller. A nice characteristic of the induction motor is that its currents and flux state–space equations can be rewritten in an LPV form with rotor speed as a scheduling variable. This feature will be exploited in designing a robust HN LPV feedback controller for the inner current loop. Also, the LPV structure of the motor can be used to prove the robustness of the flux observer and to compute the worst case flux estimation error in terms of the HN norm, with respect to rotor speed and rotor resistance variations. Two LPV flux estimators are proposed. The first is a simple LPV reference model. However, analysis of the flux estimation error reveals that the reference model is too sensitive with respect to parameter variations to provide good flux tracking. Therefore, a second LPV observer using the stator currents and stator voltages is proposed to improve the robustness of the flux estimation. The application is based on an experimental set-up available at the Laboratory of Electrical Engineering, Paris. The non-linear model of the motor and tracking requirements come from a recent benchmark (Ortega, Ashen, & Mendes, 2000). Full non-linear simulations, including parametric variations, saturation limits, time delay and highly noisy measurements demonstrate that modern HN techniques combined with input/output linearization offer a promising and effective way to design robust controllers for induction motors. The paper is organized as follows. The model of the induction motor is given in Section 2. The controller design is described in Section 3. Section 4 includes the simulation results and conclusions are given in Section 5.

2. Induction motor model A squirrel-cage induction motor is considered whose nominal physical parameters are given in Table 1. Under the assumption of linear magnetic circuits, a 5th order non-linear model (stator-fixed frame) of the induction motor is given by 8 x’ ¼ a1 ðx2 x5  x3 x4 Þ þ a2 x1 þ a3 tL ; > > > 1 > > x’ 2 ¼ a4 x2  np x1 x3 þ a5 x4 ; > > > > < x’ 3 ¼ np x1 x2 þ a4 x3 þ a5 x5 ; ð1Þ G: > x’ 4 ¼ a6 x2 þ a7 x1 x3  gx4 þ a8 u1 ; > > > > > x’ 5 ¼ a7 x1 x2 þ a6 x3  gx5 þ a8 u2 ; > > > : y ¼ ½x1 ; x4 ; x5 T : The state vector is x ¼ ½o; fa ; fb ; ia ; ib T ¼ T ½x1 ; x2 ; x3 ; x4 ; x5 ; where o is the rotor speed, f ¼ ½fa ; fb T are the rotor fluxes, is ¼ ½ia ; ib T are the stator currents and us ¼ ½u1 ; u2 T represents the stator voltages.

Table 1 Nominal physical parameters of the induction motor Description

Parameter

value

Units

Stator Inductance Rotor Inductance Mutual Inductance Leakage factor ss ¼ sr Stator resistance Rotor resistance Moment of inertia Viscous damping constant Number of pole pairs

Ls Lr Lsr s Rs R% r Dm Rm np

0.47 0.47 0.44 0.12 0.8 3.6 0.06 0.04 2

H H H O O kg m2 Nms

The measured output is y ¼ ½o; ia ; ib T ; the control input is us ¼ ½u1 ; u2 T and tL is the load torque disturbance. The outputs to be controlled are * *

Shaft rotor speed x1 ¼ o: Rotor flux norm F ¼ jjfjj:

For ease of notation, the same letter G will be used in block diagrams to denote sub-parts of the nominal model of the induction motor; in some diagrams therefore the input/output vector map differs from (1). Remark. The flux F has to be controlled but it is not available for measurement. This makes the tracking of the flux especially difficult since the robustness of the tracking relies entirely on the accuracy of the flux estimation. The parameters of the induction motor model are defined as follows: a1 ¼ np Lsr =ðDm Lr Þ; a2 ¼ Rm =Dm ; a3 ¼ 1=Dm ; a4 ¼ 1=Tr ; a5 ¼ Lsr =Tr ; a6 ¼ Lsr =ðTr sLs Lr Þ; a7 ¼ np Lsr =ðsLs Lr Þ and a8 ¼ 1=ðsLs Þ where Tr ¼

Lr ; Rr

s ¼1 

g¼

L2sr Ls Lr

Rs L2sr þ ; Ls s Ls sLr Tr

ð2Þ

with nominal values given in Table 2. All the parameters are known except the rotor resistance Rr ; which will change in the experiments, and the load torque disturbance tL : Rr is assumed to vary in the range ½0:7R% r ; 1:3R% r : The change in the rotor resistance affects linearly or affinely the values of parameters a4 ; a5 ; a6 and g: The load torque disturbance amplitude is unknown but is assumed to vary in the range ½0:25t% L ; t% L : In Table 2, the nominal values of parameters subject to variation are denoted by a% 4 ; a% 5 etcy; similarly, R% r in Table 1.

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Table 2 Nominal parameter values of the induction motor model Parameter

Nominal value

a1 a2 a3 a% 4 a% 5 a% 6 a7 a8 g% t% L

31.21 0.667 16.67 7.66 3.37 127.14 33.19 17.73 197.78 7 (Nm)

Fig. 1. Controller structure.

cated two-degree-of-freedom controller, performance should be substantially easier to prove. 3. Controller structure and controller design 3.1. Polytopic HN current feedback In this section, we will introduce and discuss the advantages and the shortcomings of the controller structure. The controller is made up of four elements which will be designed separately. Fig. 1 shows the structure considered. Klpv is a current feedback controller which tracks the current setpoint reference isref : Its input is the difference between isref and the stator current vector is and it is scheduled with the rotor speed o: Klpv is designed from the state–space model governing the flux and current vectors which turns out to be an affine parameter dependent model when the rotor speed o is taken as the scheduling variable. The role of Klpv is to ensure good tracking over the entire operating range (i.e. when o is varying). It will have an LPV form and be designed using quadratic HN gain scheduling stabilization (Apkarian & Gahinet, 1995; Apkarian & Adams, 1998; Gahinet et al., 1995). The reference current vector isref is generated through a static input–output linearization state feedback Fio : Fio is used to linearize and decouple the flux and the rotor speed from the input n shown in Fig. 1. The flux vector is estimated by Gf (and Gobs in the sequel) which, as we will see, will also have an LPV structure. Finally, Klin is a simple LTI regulator based on the linearized map relating the input n with the speed and the flux. Klin will be a static diagonal controller adjusted to ensure robust tracking of the speed and of the flux. A major advantage of this control structure is that the overall control problem is simplified by subdividing it in four (fairly) independent sub-control problems. This allows us to take advantage of the LPV description of the motor both in the current feedback and the flux observer designs. Also, the resulting controller is relatively simple and easy to implement. However, it turns out that stability and performance of the closedloop cannot be straightforwardly established even in the nominal case. Robust stability analysis is beyond the scope of this paper. However, using a more sophisti-

In this section, attention is focused on the design of the current feedback controller. Let us consider the second, third, fourth and fifth state space equations of the induction motor given in (1). These four equations constitute an affine parameter-dependent plant if the rotor speed x1 ¼ o is taken as the parameter. More precisely, let us define the subsystem G1 with state vector x ¼ ½x2 ; x3 ; x4 ; x5 T having is ¼ ½x4 ; x5 T as output vector and us ¼ ½u1 ; u2 T as input vector. From (1) G1 can be written as follows: ( x’ ¼ ðA0 þ oA1 Þx þ Bu; G1 : ð3Þ is ¼ Cx with 0

a4 B0 B A0 ¼ B @ a6 0

0 a4

a5 0

0 a6

g 0

0

0 B n B p A1 ¼ B @ 0 a7 0

1 0 a5 C C C; 0 A g

np 0

0 0

a7

0

1 0 0C C C; 0A

0

0

0

0

0

1

B0 B B¼B @ a8

0 0

C C C; A

0

a8

ð4Þ

C¼

0 0

ð5Þ

0 0

1 0

0 1

! ð6Þ

and with oA½omin ; omax ¼ ½110; 110 rad=s:

ð7Þ

Alternatively, G1 ðtÞ admits the following polytopic state–space representation: G1 : aðtÞS1 þ ð1  aðtÞÞS0

ð8Þ

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with S0 ¼ ðA*0 ; B; C; 02
2 Þ; S1 ¼ ðA*1 ; B; C; 02
2 Þ; A*0 ¼ A0 þ omin A1 ; A*1 ¼ A0 þ omax A1 ; oðtÞ  omin 0paðtÞ ¼ p1: omax  omin

Fig. 3. Current feedback.

ð9Þ

Due to the symmetry of the model in rotor speed o and because omin ¼ omax ; it is easy to verify, using the state coordinate transformation with the permutation matrix 0 1 0 1 0 0 B1 0 0 0C B C ð10Þ T ¼ T 1 ¼ B C @0 0 0 1A 0

0

1

0

N

that the two systems S0 and S1 (at the vertices) have the same evolution matrix, and thus the same dynamics. They differ only by a permutation of their input/output maps since S* 1 ¼ ðA*0 ; TB; CT; 02
2 Þ:

ð11Þ

Therefore, the subsystem G1 with polytopic representation ðS* 1 ; S0 Þ is quadratically stable since there exists a common Lyapunov function which proves the stability of S* 1 and S0 : Step responses of G1 computed for various value of o are given in Fig. 2. Clearly, the induction motor exhibits significant cross-couplings and badly damped modes whose frequencies vary considerably with the rotor speed. The robust multivariable LPV controller Klpv has to provide satisfactory performance over the entire operating range of the motor (i.e. when o varies). The design problem is tackled using the one-degree-of-freedom 0.16 0.14 0.12 0.1

i s (A)

0.08 0.06 0.04 0.02 0 _ 0.02 _ 0.04

0

0.05

0.1

0.15

0.2

control structure depicted in Fig. 3. A mixed-sensitivity T=S loop shaping quadratic-gain HN optimization is proposed for the design of the polytopic regulator Klpv : The optimization problem is to find a stabilizing control law Klpv to minimize, for all o in (7), in a quadratic HN sense, the cost function " #  W T  T   ð12Þ   :   WS S  

0.25 time (s)

0.3

0.35

0.4

0.45

0.5

Fig. 2. Open-loop time responses of G1 : Unit step demand in u1 ; for 20 equally spaced values of o in the range (7).

WS and WT are used to shape the output sensitivity function S ¼ ðI þ G1 Klpv Þ1 and the complementarity sensitivity function T :¼ I  S: For good tracking accuracy, the sensitivity function is required to be small. A high gain low-pass filter can be used to give accurate tracking and for our problem, the following weight was selected: WS ¼ diagðwS ; wS Þ; wS ¼

s þ oBS AS 1=MS s þ oBS

ð13Þ

with AS ¼ 1=500; MS ¼ 2; oBS ¼ 550 rad=s: WS specifies a disturbance attenuation of 1=A at low frequencies, a maximum amplification of 1=MS and a bandwidth which is approximately oBS (see Fig. 4). In this application, it was not found necessary to use a dynamic weight to shape the complementary sensitivity and the following constant weight was chosen: WT ¼ diagð0:8; 0:8Þ:

ð14Þ

The polytopic regulator Klpv was computed using the function hinfgs in the LMI control toolbox which makes use of a single Lyapunov function to compute a polytopic HN output feedback synthesis (Gahinet et al., 1995). With the weighting functions defined above, the LMI optimization gave a polytopic regulator with two vertices, each vertex being an LTI regulator with six states. Four states come from the plant and two from WS : The HN performance, g1 ; was approximately 1.37. The closed-loop sensitivity functions, for 10 equally spaced values of o in the range (7), are given in Fig. 4. The performance requirements specified by WS and WT are met for any value of the shaft rotor speed o: From, a robustness point of view, we can mention that the maximum peak on the complementary sensitivity function is 0:1 dB and the maximum amplification on the sensitivity function S less than 1:2: This ensures excellent phase and gain margins at the plant output. The high

E. Prempain et al. / Control Engineering Practice 10 (2002) 633–644 1

637

1

10

10

0

Magnitude (T)

Magnitude (S)

10

0

10

_1

10

_2

10

_1

_3

10

3

2

10

4

10 Frequency [rad/s]

10

10

10

3

2

4

10 Frequency [rad/s]

10

Fig. 4. Singular value plots of the current feedback: sensitivity at the plant output S and inverse of performance weight WS ; complementary sensitivity T and inverse of performance weight WT ; for 10 equally spaced values of o in the range (7).

30

1.4

1.2

25

1

20

0.8 us (V)

is (A)

15 0.6

10 0.4 5 0.2 0

0

_ 0.2

0

1

2

3

4

5 time ms

6

7

8

9

10

_5

0

1

2

3

4

5

6

7

8

9

10

time ms

Fig. 5. Unit step closed-loop responses of the current gain scheduled feedback system along the rotor speed trajectory given in (15). Stator currents (left) and corresponding stator tensions (right).

frequency roll-off of T is satisfactory. Good tracking is expected at frequencies less than 400 rad=sE64 Hz: Note that the frequency responses of S and T are oindependent. This can be explained by the symmetry of the model in o: Recall that the polytopic system given in (9) is made up of S0 and S1 which are equivalent dynamical systems (actually, they represent the same LTI system after permutating their inputs and outputs). The closed-loop time-responses of the current controlled system for a 1 A step demand in ia along the rotor speed trajectory oðtÞ ¼ 1000t

ð15Þ

are given in Fig. 5. The decoupling is excellent. The settling time is about 60 ms and the overshoot of 1:1 A is small. The overshoot on the control input is about 27 V which can be considered satisfactory.

3.2. Input–ouput linearization qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Let Y ¼ ½o; F with F ¼ f2a þ f2b : The basic idea of the input–output linearization is to produce a linear differential relation between the output Y and a new input n: This can be done by simply differentiating the output vector Y repeatedly until the control input appears e.g. (Slotine & Li, 1991). For the induction motor, the relative degree is well defined and the zero dynamics are stable; thus the input–output linearization can be performed. Let us consider the three first equations of (1). Assuming that the rotor speed and the flux vector f are available for measurement, our aim is to construct a non-linear relation to produce an ideal reference stator current vector isref to make the closedloop input–output map Tn;Y linear. Let n2 be the first derivative of the flux. From (1) and using the definition

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of F; we get ’ ¼ fa ða% 4 fa þ a% 5 iaref Þ þ fb ða% 4 fb þ a% 5 ibref Þ :¼ n2 : F F

ð16Þ

Let n1 be the first derivative of the rotor speed o ¼ x1 n1 is given by o ’ ¼ a1 ðfa ibref  fb iaref Þ þ a2 o þ a3 t% L :¼ n1 : Eqs. (16) and (17) can be rewritten as " #" # " # iaref Fðn2  a% 4 FÞ fa a% 5 fb a% 5 ¼ : n1  a3 t% L  a2 o a1 fb a1 fa ibref Thus,

"

Fio ðo; f; nÞ :¼

iaref

ð17Þ

ð18Þ

#

ibref

" a1 fa 1 ¼ 2 a1 a% 5 F a1 fb

fb a% 5 fa a% 5

#"

# Fðn2  a% 4 FÞ : n1  a3 t% L  a2 o

ð19Þ The non-linear state feedback control law isref ¼ ½iaref ; ibref T given in (19) is well defined whenever Fa0 and results in 8 ’ ¼ n1 ; > : Y ¼ ½o; F : (Fig. 6). 3.3. Reference model The input–output linearization assumes that the fluxes are measurable. However, in practice only the stator currents and the rotor speed are available for control purposes. To estimate the fluxes, it is proposed first to use a reference model which is a copy of the second and the third state equations of the induction motor given in (1). Let a% 4 (resp a% 5 ) be the nominal value of a4 (resp a5 ) and let Gf be the reference model governed by 8 < f’# a ¼ a% 4 f # a  np of # b þ a% 5 ia ; Gf : ð21Þ : f’# ¼ n of# þ a f % 4 # b þ a% 5 ib p a b

Recall that G1 given in (3) admits a quadratically stabilizable polytopic representation and so does Gf : The interconnection structure (S) shown in Fig. 7 represents the difference (ef ) between the estimated and real flux vectors. To estimate the robustness of the estimation with respect to speed and rotor resistance variations one can compute the quadratic HN norm of S i.e. the smallest g2 > 0 such that jjef jjL2 pg2 jjis jjL2 : Clearly, a small value of the HN norm of S :¼ Tef ;is over the speed range o and for any possible value of a4 will indicate a good flux estimation. The interconnected system of Fig. 7 is governed by " # " #" # 8 f a4 I2 þ oAf 02
2 d f > > > ¼ > > # dt f# a% 4 I2 þ oAf 02
2 > f > > > " # > < a4 I 2 ð24Þ S: Lsr is ; > a% 4 I2 > > > " # > > >   f > > > : : ef :¼ f  f# ¼ I2  I2 f# Now, let us consider S as an affine dependent model with parameters: 110pop110

 9:96pa4 p  5:36:

ð25Þ

The quadratic HN norm of S is g2 ¼ 0:143: g2 is an upper bound on the worst-case amplification error for unbounded rates in the variations of o and a4 : Assume that jjis jjo12 A then the worst-case flux estimation error will be in any case less than 0:14
12 ¼ 1:68 Wb: This is a rather pessimistic estimate since it does not take into

Fig. 6. Input–Ouput linearization.

with inputs ½o; ia ; ib : The outputs of Gf are the # a and f# b : (21) can be rewritten in a estimated fluxes f more compact form as ’# ¼ a I þ oA  L a i : f ð22Þ %4 2 f sr % 4 s with Af ¼

"

0 np

np 0

#

and f ¼ ½fa ; fb T and is ¼ ½ia ; ib T :

ð23Þ Fig. 7. Interconnection structure representative of the system S: G is the induction motor with the uncertain parameter a4 and Gf is the reference model.

E. Prempain et al. / Control Engineering Practice 10 (2002) 633–644

flux observer. Let us consider the block diagram of Fig. 9. The matrix dI2 represents the rotor resistance uncertainty. It is normalized in such a way that jjdjjN p1: More precisely, the uncertain parameter a4 can be rewritten as a4 ¼ a% 4 ð1 þ da Þ; jda jodmax ; dmax ¼ ða4max  a4min Þ=2a% 4 : Let us consider, the following LTI system which corresponds to S0 (the vertex for o ¼ omin ). 8 > < x’ ¼ ða% 4 A2 þ omin A1 Þx þ P12 ud =dmax þ Bu; ð26Þ S0d : yd ¼ P21 x; > : y¼x

0

10

_1

Magnitude (Σ)

10

_2

10

with _3

10

639

0

10

1

2

10

10

3

10

Frequency [rad/s]

Fig. 8. Singular value plot of S (scaled) for 5 and 4 equally spaced values of o and a4 in the ranges (25), respectively.

account the rate of variation of the parameters, the directionality of the current vector and the peak frequency where this worst-case error occurs. However, it indicates that the reference model may not be appropriate in the presence of parametric uncertainty. It is also instructive to plot the frequency response of S for various values o and a4 : Fig. 8 shows the singular value plots of S (scaled by 12I2 ; the maximum expected current amplitude). The steady-state offset is small (about 0.3). However, S has badly damped modes situated between 40 and 200 rad=s: This may lead to oscillatory transient responses for the flux estimation. Simulation results presented in the next section, will reveal that the estimation provided by Gf is sufficient to ensure robust stability despite variation in the rotor resistance. However, the HN gain of S is not small enough to guarantee good flux tracking for some significant changes in a4 (i.e. the rotor resistance). 3.4. LPV flux observer design A better LPV flux estimator can be designed using both tension and current measurements. The design is based on the plant G3 which is equal to G1 but with C ¼ I4 ; where a4 is the uncertain parameter which is linearly dependent on the rotor resistance. The aim is to design an observer able to maintain a good flux estimation at any speed despite uncertainty in the rotor resistance. We know that G1 admits a polytopic representation ðS0 ; S1 Þ where a4 enters linearly in S0 and S1 : Therefore, we can extract a4 from the polytopic representation using the concept of linear fractional transformations (see e.g. Zhou, Doyle, & Glover, 1995). Our aim is to take explicitly into account the uncertainty due to the rotor resistance variation in the design of the

0

B B 0 B A2 ¼ B Lsr B sLs Lr @ 0 P12 ¼

P21 ¼

0

Lsr

1

0

0

L2sr sLs Lr

sLLssrLr

0

1

1

0

sLLssrLr

0

0

1

0

sLLssrLr ! 0

1

0

Lsr

0

1

0

0

1

C Lsr C C C; 0 C A

ð27Þ

L2sr sLs Lr

!T ;

Lsr

ð28Þ

ð29Þ

and where A1 and B are given in (5) and (6), respectively. It is easy to show that S0 ¼ Fl ðdI2 ; S0d Þ

ð30Þ

and S1 ¼ Fl ðdI2 ; S1d Þ;

ð31Þ

where S1d is the same as S0d if one replaces omin by omax in (26) and where d :¼ da =dmax : The observer design is based on the interconnection of Fig. 9. The regulated output is the weighted flux estimation error ef : The measured outputs are the current vector and the supply voltage. Set " # " # " # ud yd us # w¼ ; z¼ ; u ¼ f; y ¼ : ð32Þ us ef is

Fig. 9. Flux observer interconnection structure.

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The observer design consists of finding u ¼ Gobs y to minimize, for all admissible speed trajectories, the closed-loop quadratic HN performance from w to z (g3 ). According to the small gain theorem, robust stability and robust performance will be achieved whenever g3 o1: As is customary in m-synthesis (e.g. Zhou et al., 1995; Balas et al., 1993; Boyd, El Ghaoui, Feron, & Balakrishnan, 1994) we can enforce a better minimization of Tef ;us ; by scaling ud by ds ¼ diagðd1 ; d2 Þ with d1 > 0; d2 > 0 and yd by ds1 without affecting performance and stability. Wf was chosen as the identity matrix and after some trial and error ds was selected as ds ¼ diagð3:7; 3:7Þ:

ð33Þ

The polytopic observer Gobs was computed using the function hinfgs in the LMI control toolbox (Gahinet et al., 1995). With the weighting functions defined above, the LMI optimization gave a polytopic observer with two vertices, each vertex being an LTI regulator with four states. The HN performance, g3 ; was approximately 1.17. It is easy to check that jjTyd ;ud jjN o1: Therefore, the observer is robustly, quadratically stable (i.e. stable for any variation of the rotor resistance and for any speed trajectory). Since g3 E1:17; there is a worst-case for which the flux estimation error can be about 1:17: This is again a very pessimistic performance estimate. But we have to keep in mind that this estimate is given for unbounded variations in the rotor speed and for any bounded d (possibly non-linear) with L2 -norm less than dmax : It is also instructive to compute the frequency response of Tef ;us for various values o and a4 as given in Fig. 10. In order to compare them with the singular value plots of the reference model and the LPV observer, Tef ;us ; us has been scaled by 210 (the maximum

tension amplitude). The plots of Figs. 8 and 10 correspond to the same combination of speed and rotor resistance. Clearly, the LPV observer, provides a better flux estimation than the reference model since the steady state error is about 0.1 and the maximum amplification error less than 0.3. 3.5. Control of the speed and the flux To control the speed and the flux, one has to consider the plant G2 given in (20). It is a chain of two integrators, representing the nominal closed-loop input–output map TY ;n when the condition is ¼ isref holds. The regulator can be designed using any classical linear method. In this example, a simple diagonal gain was found to provide good tracking and sufficient robustness. Trial and error led to Klin ¼ diagð343; 286Þ; for which the closed-loop bandwidth is approximately 300 rad=s: This was about the largest bandwidth we thought practical. Instability due to unmodelled high frequency dynamics (time-delays which are present in the real closed-loop) or due to inaccurate flux estimation (making apparent the non-linearities of TY ;n ) may occur for higher gains. 3.6. Summary of the design process *

0

10

_ 10 1

Magnitude

_ 10 2

_ 10 3

*

_ 10 4

_ 10 5

_ 10 6 _ 2 10

10

_1

10

0

1

10 Frequency [rad/s]

10

2

10

3

10

4

Fig. 10. Singular value plot of Tef ;us for 5 and 4 equally spaced values of o and a4 in the ranges (25), respectively.

Current LPV feedback loop. The design is fairly systematic. It is based on the partial LPV description of the motor given in (3). An S=T design is proposed in the paper while some other forms of augmentations could be used if desired. WS should be selected so that the singular values of the sensitivity function are desirable. Typically WS has high gain at low frequencies and a gain less than one at high frequencies. Bandwidth and additional roll-off at high frequencies are usually imposed by WT which is smaller than one at low frequencies and large at high frequencies. Here there is a trade-off between tracking accuracy and robustness. Some trial and error modifications of WS and WT are necessary to adjust the bandwidth of the current loop. In this application a bandwidth of about 550 rad=s seems to be sufficient. A higher bandwidth would have resulted in a very sensitive (not robust) controller to neglected dynamics (e.g. unmodelled small time delays) and to noise measurement. Flux observer. As demonstrated in the paper, a reference model can be used to estimate the flux. However, better results are obtained with an LPV observer. The LPV observer synthesis is based on the interconnection structure of Fig. 9. The accuracy of the estimation is determined by the weighting function Wf which could be chosen as a high gain low pass filter. Because the rotor resistance variation is taken into account as a parametric uncertainty, one

E. Prempain et al. / Control Engineering Practice 10 (2002) 633–644

*

τ L (Nm)

3

2.5

2

1.5

4. Simulation results

0

1

2

3

4

5 time (s)

6

7

8

9

10

0

1

2

3

4

5 time (s)

6

7

8

9

10

5 4.5 4

Rr (Ω)

*

3.5

3.5 3 2.5

Fig. 11. Load torque and rotor resistance variations.

120 100 80

omega rad/s

*

can reduce the conservatism of the HN optimization by scaling the signals ud and yd by d ¼ diagðd1 ; d2 Þ and d 1 ; respectively. Finding d involves some trial and error. Input–Output feedback linearization. The non-linear state feedback which is used to generate the proper current signals (with respect to the speed and flux demands) is given in (19). This step does not require any tuning. Speed and flux control. This is just a matter of designing a feedback regulator Klin for a chain of two integrators (the linearized speed and flux). Note that the two-integrator model is an ideal model (assuming perfect flux measurement, perfect tracking of the currents and no uncertainties). Discrepancies between this ideal model and the actual system exists. Therefore, the feedback controller Klin has to be designed to get a reasonable bandwidth (not too high). Klin can be chosen as a constant diagonal gain. Finally, the overall controller is formed according to the interconnection given in (Fig. 1).

641

60 40 20 0

jisrefi jo7 A;

i ¼ 1; 2

ð34Þ

and for the control signal jui jo210 V;

i ¼ 1; 2:

ð35Þ

The flux and speed demands were filtered through the following second order filters Fref :¼

o2n s2 þ 2xon s þ o2n

ð36Þ

with on ¼ 8; x ¼ 0:8 to penalize the control effort. 4.1. Controller using the reference model The time responses of the closed-loop system are given in Fig. 12. The rotor speed response follows the specified reference accurately despite changes in the

1

2

3

4

5 time sec

6

7

8

9

10

1

2

3

4

5 time sec

6

7

8

9

10

1.4 1.2 1 phi (Wb)

The performance of the controller is investigated by simulations. The initial state of the motor is zero. Full non-linear simulations were carried out for the speed and flux step demand profiles and for parameter variations shown in Fig. 11. Fig. 11 represents the worst-case rotor resistance changes which may correspond to breaks in the rotor bars. Saturation limits on the stator tensions and on the reference currents were included to prevent peaks especially at initialization. Zero-order holds with a sampling frequency of Fs ¼ 4 kHz and a unit time delay Ts ¼ 1=Fs were added in the feedback loop to simulate a digital implementation. The saturation limits on the reference current components were taken as

_ 20 0

0.8 0.6 0.4 0.2 0 0

Fig. 12. Speed and flux tracking with the reference model Gf :

parameters (rotor resistance and load torque) and also despite saturation in the stator voltage occurring between t ¼ 2 and 3 s: The situation, however, is different for the tracking of the flux which depends entirely on the accuracy of the flux estimation. As predicted by the frequency analysis given in the previous section, tracking of the flux is deficient with rotor resistance changes. 4.2. Controller using the LVP observer The estimation of the flux is provided by the LPV observer. The response of the rotor speed, given in Fig. 13, is similar to the one given in Fig. 12. But, the tracking of the flux is significantly better with the LPV observer. Figs. 14 and 15 show that the modulus of the control signal (us ) stays within the limits except between t ¼ 2 and 3 s when the speed demand is 110 rad=s: The

E. Prempain et al. / Control Engineering Practice 10 (2002) 633–644

642 120 100 omega rad/s

80 60 40 20 0 _ 20

0

1

2

3

4

5 time sec

6

7

8

9

10

0

1

2

3

4

5 time sec

6

7

8

9

10

1.4 1.2

phi (Wb)

1 0.8 0.6 0.4 0.2 0

Fig. 13. Speed and flux tracking with the LPV observer Gobs :

300

ua, ub (V)

200 100 0 _ 100 _ 200 _ 300

0

1

2

3

4

5 time sec

6

7

8

9

stator current modulus remains below the limit of 12 A (Fig. 15). The simulation results presented here are very similar, in terms of tracking responses, to the sliding mode controller responses proposed in Benchaib and Edwards (2000). For the sliding mode controller, the tracking of the flux is a little bit more accurate with respect to stator voltage saturation occurring between t ¼ 2 and 3 s: However, our speed responses are a little bit faster. Also, the LPV-based controller constrasts with the controller given in Benchaib and Edwards (2000) by producing naturally smoother control signals with lower bandwidth (no chattering) and there is no saturation in the control signal for nominal speed demands (see Fig. 15). The simulation results presented here would appear to be better than those given in Ding, Wang, and Han (2000) which uses HN design techniques with linearizing state feedback and a reference model to provide the flux. Nevertheless, their results are difficult to compare in detail because only speed and flux tracking errors are given and these without the corresponding reference demands and stator tensions. In Ding et al., (2000) the speed error can be up to 6 rad=s and the flux error twice as large as it is with this design. Stator current demands are also higher.

10

4.3. Controller using the LVP observer with noisy measurements

10

ia, ib, (A)

5

0 _5 _ 10

0

1

2

3

4

5 time sec

6

7

8

9

10

Fig. 14. Stator voltage us and stator current is (LPV observer).

It is also important to study robustness of the closedloop when the current and speed measurements are corrupted by noise. For the simulation, the speed channel is corrupted with a white noise with maximum amplitude of 1 rad=s and a significant white noise level of 0:5 A is added to the stator current channels. Figs. 16–18 show the closed-loop time responses in the presence of noisy measurements. The speed response is

12 120

10

100

6 4 2 0 0

80

omega rad/s

Is (A)

8

60 40 20

1

2

3

4

5 time sec

6

7

8

9

10

0

_ 20

0

1

2

3

4

5 time sec

6

7

8

9

10

1

2

3

4

5 time sec

6

7

8

9

10

300 2

250

1.5

150

phi (Wb)

Us (V)

200

100 50 0 0

1

0.5 1

2

3

4

5 time sec

6

7

8

9

10

Fig. 15. Stator voltage modulus jjus jj and stator current modulus jjis jj with limits (LPV observer).

0 0

Fig. 16. Speed and flux tracking (LPV observer, noisy measurements).

E. Prempain et al. / Control Engineering Practice 10 (2002) 633–644 300 200 ua, ub (V)

100 0 _ 100 _ 200 _ 300

0

1

2

3

4

5 time sec

6

7

8

9

10

1

2

3

4

5 time sec

6

7

8

9

10

15

ia, ib, (A)

10 5 0 _5 _ 10 _ 15

0

Fig. 17. Stator voltage us and stator current is ; (LPV observer and noise measurement). 14 12

Is (A)

10 8 6 4 2 0

0

1

2

3

4

5 time sec

6

7

8

9

10

0

1

2

3

4

5 time sec

6

7

8

9

10

300 250

Us (V)

200 150 100 50 0

643

only the stator currents and the rotor speed were available for measurement. The controller has a structure made up of four different sub-systems: an LPV feedback current loop, a non-linear linearizing state feedback, an LPV observer and an LTI regulator. An LMI-based approach, has been proposed to design a quadratically stable flux observer and an output feedback regulator to track the stator currents. In both cases we have obtained a scheduled time varying system (LPV) which ensures a finite L2 attenuation for a given closed-loop transfer function which represents the design requirements (tracking robustness for the LPV current regulator and flux estimation error for the LPV observer). The main advantage of using LPV methods is that they provide a systematic way of designing an HN flux observer for the induction motor assuming that the rotor speed is available. Stability of the flux estimator was demonstrated using small-gain based analysis. The results for the benchmark were found to be satisfactory even in the presence of significantly noisy measurements. Due to the very low order of the control system’s components (Klpv is a 2 by 2 speed dependent regulator with only six states, the speed dependent flux observer has only four states, four inputs and two outputs) the proposed control law is easy to implement and can work with relatively low cost DSP cards. A drawback of the proposed controller is that it requires a measure of the rotor speed; first for the tracking of rotor speed, and second to update the LPV parts of the control law. Further research is required to demonstrate the robustness of the stability of the overall closed-loop and also to extend the applicability of LPV design techniques in the absence of rotor speed measurement.

Fig. 18. Stator voltage modulus jjus jj and stator current modulus jjis jj with limits (LPV observer, noisy measurements).

Acknowledgements still excellent. The flux response is noisy but still within its specified reference for t > 3 s: Between t ¼ 0 and 1 s; the flux presents an oscillatory behaviour. The case corresponds to a low speed demand, for which currents and tensions would be relatively small for a perfect output measurement. A significant level of noise is applied, therefore the signal/noise ratio in the current channels is small. This leads to an inaccurate flux estimation which in addition corrupts the decoupling properties of the input–output linearization. For higher speed demands, when the current/noise ratio increases, the flux estimation is improved and hence also tracking of flux.

5. Conclusions This paper has presented a non-linear controller design for an induction motor. It was assumed that

The authors would like to acknowledge financial support from the UK Engineering and Physical Sciences Research Council and the anonymous reviewers for their suggestions.

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Bose, B. K. (1998). High performance control of induction motor drives. IEEE Industrial Electronics Society Newsletter, 45(3), 7–11. Boyd, S., El Ghaoui, L., Feron, E., & Balakrishnan, V. (1994). Linear matrix inequalities in system and control theory. Studies in Applied Mathematics. SIAM, Philadelphia, PA. Ding, G., Wang, X., & Han, Z. (2000). HN disturbance attenuation control of induction motor. International Journal of Adaptive Control and Signal Processing, 14(2–3), 223–244. Gahinet, P., Nemirovski, A., Laub, A. J., & Chilali, M. (1995). LMI control toolbox. Natick, MA: The Math Works.

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